Atmospheric_Dynamics_III-primitive-eq_continuity-v4

# Atmospheric_Dynamics_III-primitive-eq_continuity-v4 -...

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Primitive Equations PRIMITIVE EQUATIONS

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Definition The so called primitive equations are those that govern the evolution of the large-scale motions. In other words, are the equations that describe the horizontal and vertical movement of the atmosphere and changes in temperature They are easiest to interpret when we transform the z coordinate into p coordinate
Vertical movement in p- coordinates The vertical velocity component in (x,y,p) coordinate is z p w p t p dt dp + + = V ϖ V- horizontal wind Substituting ( δ p/ δ z )=- ρ g from the Hydrostatic equation: gw ρ ϖ - 2245 gw p t p ρ ϖ - + = V 10hPa/day 100hPa/day <<10hPa/day ~ 1 week for a parcel to move from the lower to the upper troposphere Note that w and ω have opposite sign: ascending (descending) movements ω negative (positive)

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How to interpret ω Pressure ω = d p / d t 1000mb 900mb 800mb 700mb 600mb
Comparing w with ω 100hPa/day is equivalent to 1km/day or 1cm/s in the lower troposphere and twice that value in the midtroposphere (see the example shown before) (the distance between two pressure levels increases with height)

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Hydrostatic balance We saw before (Joel’s classes) that the vertical component of the movement could be described as: z z F C g z p dt dw + + - - = ρ 1 Where Cz are the vertical components of the Coriolis and Frictional forces, respectively Vertical velocities are very small and we can assume to within ~1% that the upward gradient force balances the downward pull of gravity also for large-scale motions (this approach is not valid for cloud-scale motions though). The Hydrostatic Balance can be assumed
Thermodynamic Energy Equation The evolution of the weather systems is governed by dynamical (Newton’s Laws) AND THERMODYNAMIC PROCESSES (First and second law of thermodynamics) The first law of the Thermodynamics (which represents changes and heat, expansion/contraction, increase/decrease in temperature, etc) is a prognostic equation for the parcel of air moving in the atmosphere

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First Law of Thermodynamics The first Law of the Thermodynamics can be written as: dp dT c dt J p α - = Where J represents the DIABATIC HEATING RATE (Joules kg-1 s-1 ) and dt is the infinitesimal time interval. Dividing by dt and rearranging the terms we obtain: J dt dp dt dT c p + = α Using the state equation for a substitution of α and replacing / dp dt by ω we obtain the thermodynamic energy equation p c J p T dt dT + = ϖ κ κ = R/cp=0.286
Interpretations (1) Rate of change of temperature due to ADIABATIC EXPANSION OR COMPRESSION .

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